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Posted 3 months ago
Heba was asked to find this integral using uu-substitution:
\int(-12 x-1) \sqrt{-6 x^{2}-x+1} d x

How should Heba define uu ?

Choose 1 answer:
(A) u=6x2x+1u=-6 x^{2}-x+1
(B) u=6x2x+1u=\sqrt{-6 x^{2}-x+1}
(c) u=12x1u=-12 x-1
(D) u=xu=\sqrt{x}
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Answer from Sia
Posted 3 months ago
Solution by Steps
step 2
Looking at the integrand (12x1)6x2x+1(-12x-1)\sqrt{-6x^2-x+1}, we notice that the derivative of 6x2x+1-6x^2-x+1 is 12x1-12x-1, which is present outside the square root
step 3
Therefore, we should choose uu to be the expression inside the square root, u=6x2x+1u = -6x^2-x+1, so that dudu will be proportional to 12x1dx-12x-1 \, dx
step 4
This choice will simplify the integral, as the substitution will cancel out the 12x1-12x-1 term, leaving us with an integral in terms of uu
Key Concept
In uu-substitution, we look for a function within the integral whose derivative is also present. This allows us to simplify the integral by substituting dudu for the corresponding terms in dxdx.

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